Probability, Random Variables, and Probability Distributions
When using a simulation to estimate the probability that at least 60% of a school's students will pass an AP exam, which technique most accurately reflects the complexity of student performance?
Using a simple random number generator assuming each student has an equal chance of passing.
Assigning probabilities based on individual student performance data from previous years.
Flipping a coin for each student since there are two possible outcomes: pass or fail.
Drawing names from a hat and assigning pass or fail based on the order drawn.
What is typically needed to create an accurate representation of real-life events when conducting simulations?
A model based on theoretical probability
The use of biased samples
Only analytical methods without randomness
An extremely large population size
If a student uses a simulation to estimate the probability of rain on a given day, which aspect of their simulation design will most directly influence the variability of their probability estimate?
The time of day when they run the simulation.
The number of simulations they run.
The color coding scheme for sunny and rainy days in the simulation.
The type of random device they use.
When estimating probabilities using a simulation, what do we call each individual repetition of our simulated process?
Parameter
Experiment
Hypothesis test
Trial
A simulation is conducted to estimate the probability of rolling a number greater than 4 on a six-sided die. The die is rolled 500 times in the simulation, and a number greater than 4 is rolled 350 times. What is the estimated probability based on this simulation?
0.35
0.5
0.7
0.875
In statistical simulation modeling, why is it important not to introduce personal bias when deciding how many trials to run?
Personal bias might influence trial count, leading to non-representative sampling that could skew results away from true probability.
Personal biases ensure trial counts are reasonably manageable while still yielding statistically valid data.
Biased choices regarding trial counts help streamline computation time without significantly impacting simulation outcomes.
Bias towards a higher number of trials often leads to more precise simulations as it includes a wide range of conditions.
When running multiple trials for estimation via simulation, why is it important for all trials to remain independent?
To prevent respondents from providing similar answers on surveys or experiments.
To increase the variability across trials, ensuring more robust results.
To ensure that the outcome of one trial doesn't influence another, keeping estimates unbiased.
To determine causation between variables within the simulated model.

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When simulating the flip of a fair coin, what kind of sampling method is appropriate?
Cluster sampling
Census
Simple random sampling
Multistage sampling
When using random number tables to simulate rolling a pair of dice 100 times, what must be done to ensure accurate results?
Assign two-digit numbers ranging from 01 to 36 to cover all possible outcomes equally.
Use only odd numbers in the table since there are an odd number of total outcomes (36).
Exclude numbers above 60 as they do not correspond to any dice roll outcomes.
Repeat any number sequence that doesn't directly translate into a dice outcome.
What do we call a long run relative frequency as it approaches an expected value through many trials in a simulation?
A theoretical distribution function curve.
An inferential statistics confidence interval.
A deterministic model outcome sequence.
An empirical probability estimate.